A Cycloid is a Tautochrone

In 1659 Christiaan Huygens answered a question that came to him as he watched a swinging chandelier in a church: What curve has the property that a bead sliding along it under uniform gravity and with no friction will oscillate with a period independent of the amplitude?

The answer turned out to be the cycloid generated by a circle, as illustrated in Figure 1. Several solutions to this problem have been found; Abel’s is particularly remarkable [1].

Figure 1. The contact point C is an instantaneous center of rotation of the rigid wheel, and thus v ⊥ r.

Presented here is a very short geometrical proof of the tautochronous property of the cycloid. It is based on the fact that \(\bf{v} \perp \bf{r} \), as explained in Figure 1.1

To prove that the cycloid is a tautochrone, it suffices to show that the arclength distance \(s\) from the bottom of the cycloid behaves as a harmonic oscillator:

\[\begin{equation}
a = −ks,
\end{equation}\]

where \(a = \ddot{s}\), for some constant \(k\). (This idea, which I had learned from Henk Broer, is attributed to Lagrange.) Because \(a = 0\) when \(s = 0\), we just need to verify that

1 Incidentally, building on this fact, the line of velocity of every point on a rolling wheel (in the ground reference frame) passes through the topmost point of the wheel. A pebble stuck to the tire always aims straight at, or straight away from, the topmost point of the wheel!